Browsing by Subject "Iterative solvers"

Now showing 1 - 11 of 11

Open Access
Advanced partitioning and communication strategies for the efficient parallelization of the multilevel fast multipole algorithm
(IEEE, 2010) Ergül O.; Gürel, Levent
Large-scale electromagnetics problems can be solved efficiently with the multilevel fast multipole algorithm (MLFMA) [1], which reduces the complexity of matrix-vector multiplications required by iterative solvers from O(N 2) to O(N logN). Parallelization of MLFMA on distributed-memory architectures enables fast and accurate solutions of radiation and scattering problems discretized with millions of unknowns using limited computational resources. Recently, we developed a hierarchical partitioning strategy [2], which provides an efficient parallelization of MLFMA, allowing for the solution of very large problems involving hundreds of millions of unknowns. In this strategy, both clusters (sub-domains) of the multilevel tree structure and their samples are partitioned among processors, which leads to improved load-balancing. We also show that communications between processors are reduced and the communication time is shortened, compared to previous parallelization strategies in the literature. On the other hand, improved partitioning of the tree structure complicates the arrangement of communications between processors. In this paper, we discuss communications in detail when MLFMA is parallelized using the hierarchical partitioning strategy. We present well-organized arrangements of communications in order to maximize the efficiency offered by the improved partitioning. We demonstrate the effectiveness of the resulting parallel implementation on a very large scattering problem involving a conducting sphere discretized with 375 million unknowns. ©2010 IEEE.
Open Access
Computational analysis of complicated metamaterial structures using MLFMA and nested preconditioners
(IEEE, 2007-11) Ergül, Özgür; Malas, Tahir; Yavuz, Ç; Ünal, Alper; Gürel, Levent
We consider accurate solution of scattering problems involving complicated metamaterial (MM) structures consisting of thin wires and split-ring resonators. The scattering problems are formulated by the electric-field integral equation (EFIE) discretized with the Rao-Wilton- Glisson basis functions defined on planar triangles. The resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications that are required by the iterative solvers are accelerated with the multilevel fast multipole algorithm (MLFMA). Since EFIE usually produces matrix equations that are ill-conditioned and difficult to solve iteratively, we employ nested preconditioners to achieve rapid convergence of the iterative solutions. To further accelerate the simulations, we parallelize our algorithm and perform the solutions on a cluster of personal computers. This way, we are able to solve problems of MMs involving thousands of unit cells.
Open Access
Efficient analysis of large phased arrays using iterative MoM with DFT-based acceleration algorithm
(John Wiley & Sons, Inc., 2003) Ertürk, V. B.; Chou, H-T.
A discrete Fourier transform (DFT)-based iterative method of moments (IMoM) algorithm is developed to provide an O(Ntot) computational complexity and memory storages for the efficient analysis of electromagnetic radiation/scattering from large phased arrays. Here, Ntot is the total number of unknowns. Numerical results for both printed and free-standing dipole arrays are presented to validate the algorithm's efficiency and accuracy.
Open Access
Efficient solution of the electric and magnetic current combined‐field integral equation with the multilevel fast multipole algorithm and block‐diagonal preconditioning
(Wiley-Blackwell Publishing, Inc., 2009-12) Ergül, Özgür; Gürel, Levent
We consider the efficient solution of electromagnetics problems involving dielectric and composite dielectric-metallic structures, formulated with the electric and magnetic current combined-field integral equation (JMCFIE). Dense matrix equations obtained from the discretization of JMCFIE with Rao-Wilton-Glisson functions are solved iteratively, where the matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm. JMCFIE usually provides well conditioned matrix equations that are easy to solve iteratively. However, iteration counts and the efficiency of solutions depend on the contrast, i.e., the relative variation of electromagnetic parameters across dielectric interfaces. Owing to the numerical imbalance of off-diagonal matrix partitions, solutions of JMCFIE become difficult with increasing contrast. We present a four-partition block-diagonal preconditioner (4PBDP), which provides efficient solutions of JMCFIE by reducing the number of iterations significantly. 4PBDP is useful, especially when the contrast increases, and the standard block-diagonal preconditioner fails to provide a rapid convergence.
Open Access
EFIE and MFIE, why the difference?
(IEEE, 2008-07) Chew W.C.; Davis, C. P.; Warnick, K. F.; Nie, Z. P.; Hu, J.; Yan, S.; Gürel, Levent
EFIE (electric field integral equation) suffers from internal resonance, and the remedy is to use MFIE (magnetic field integral equation) to come up with a CFIE (combined field integral equation) to remove the internal resonance problem. However, MFIE is fundamentally a very different integral equation from EFIE. Many questions have been raised about the differences.
Open Access
Extension of forward-backward method with DFT-based acceleration algorithm for the efficient analysis of large periodic arrays with arbitrary boundaries
(John Wiley & Sons, 2005) Civi, Ö. A.; Ertürk, V. B.; Chou, H.-T.
An extension of the discrete Fourier transform (DFT)-based forward-backward algorithm is developed using the virtual-element approach to provide a fast and accurate analysis of electromagnetic radiation/scattering from electrically large, planar, periodic, finite (phased) arrays with arbitrary boundaries. Both the computational complexity and storage requirements of this approach are O(Ntot) (Ntot is the total number of unknowns). The numerical results for both printed and freestanding dipole arrays with circular and/or elliptical boundaries are presented to validate the efficiency and accuracy of this approach.
Open Access
Minimizing communication through computational redundancy in parallel iterative solvers
(2011) Torun, Fahreddin Şükrü
Sparse matrix vector multiplication (SpMxV) of the form y = Ax is a kernel operation in iterative linear solvers used in scientific applications. In these solvers, the SpMxV operation is performed repeatedly with the same sparse matrix through iterations until convergence. Depending on the matrix and its decomposition, parallel SpMxV operation necessitates communication among processors in the parallel environment. The communication can be reduced by intelligent decomposition. However, we can further decrease the communication through data replication and redundant computation. The communication occurs due to the transfer of x-vector entries in row-parallel SpMxV computation. The input vector x of the next iteration is computed from the output vector of the current iteration through linear vector operations. Hence, a processor may compute a y-vector entry redundantly, which leads to a x-vector entry in the following iteration, instead of receiving that x-vector entry from another processor. Thus, redundant computation of that y-vector entry may lead to reduction in communication. In this thesis, we devise a directed-graph-based model that correctly captures the computation and communication pattern for above-mentioned iterative solvers. Moreover, we formulate the communication minimization by utilizing redundant computation of y-vector entries as a combinatorial problem on this directed graph model. We propose two heuristics to solve this combinatorial problem. Experimental results indicate that the communication reducing strategy by redundantly computing is promising.
Open Access
A novel method for scaling iterative solvers: avoiding latency overhead of parallel sparse-matrix vector multiplies
(Institute of Electrical and Electronics Engineers, 2015) Selvitopi, R. O.; Ozdal, M. M.; Aykanat, Cevdet
In parallel linear iterative solvers, sparse matrix vector multiplication (SpMxV) incurs irregular point-to-point (P2P) communications, whereas inner product computations incur regular collective communications. These P2P communications cause an additional synchronization point with relatively high message latency costs due to small message sizes. In these solvers, each SpMxV is usually followed by an inner product computation that involves the output vector of SpMxV. Here, we exploit this property to propose a novel parallelization method that avoids the latency costs and synchronization overhead of P2P communications. Our method involves a computational and a communication rearrangement scheme. The computational rearrangement provides an alternative method for forming input vector of SpMxV and allows P2P and collective communications to be performed in a single phase. The communication rearrangement realizes this opportunity by embedding P2P communications into global collective communication operations. The proposed method grants a certain value on the maximum number of messages communicated regardless of the sparsity pattern of the matrix. The downside, however, is the increased message volume and the negligible redundant computation. We favor reducing the message latency costs at the expense of increasing message volume. Yet, we propose two iterative-improvement-based heuristics to alleviate the increase in the volume through one-to-one task-to-processor mapping. Our experiments on two supercomputers, Cray XE6 and IBM BlueGene/Q, up to 2,048 processors show that the proposed parallelization method exhibits superior scalable performance compared to the conventional parallelization method.
Open Access
Preconditioning iterative MLFMA solutions of integral equations
(IEEE, 2010) Gürel, Levent; Malas, Tahir; Ergül, Özgür
The multilevel fast multipole algorithm (MLFMA) is a powerful method that enables iterative solutions of electromagnetics problems with low complexity. Iterative solvers, however, are not robust for three-dimensional complex reallife problems unless suitable preconditioners are used. In this paper, we present our efforts to devise effective preconditioners for MLFMA solutions of difficult electromagnetics problems involving both conductors and dielectrics. © 2010 IEEE.
Open Access
Solution of low-frequency electromagnetics problems using hierarchical matrices
(IEEE, 2013) Kazempour, Mahdi; Gürel, Levent
Fast and accurate solutions of low-frequency electromagnetics problems are obtained with an iterative solver based on hierarchical matrices. Iterative solvers require matrix-vector multiplications (MVMs). The results show significant reductions both in CPU time and memory consumption compared to the O(N2) complexity of ordinary matrix filling and MVM.
Open Access
Solutions of large-scale electromagnetics problems using an iterative inner-outer scheme with ordinary and approximate multilevel fast multipole algorithms
(2010) Ergül, A.; Malas, T.; Gürel, Levent
We present an iterative inner-outer scheme for the efficient solution of large-scale electromagnetics problems involving perfectly-conducting objects formulated with surface integral equations. Problems are solved by employing the multilevel fast multipole algorithm (MLFMA) on parallel computer systems. In order to construct a robust preconditioner, we develop an approximate MLFMA (AMLFMA) by systematically increasing the efficiency of the ordinary MLFMA. Using a flexible outer solver, iterative MLFMA solutions are accelerated via an inner iterative solver, employing AMLFMA and serving as a preconditioner to the outer solver. The resulting implementation is tested on various electromagnetics problems involving both open and closed conductors. We show that the processing time decreases significantly using the proposed method, compared to the solutions obtained with conventional preconditioners in the literature.